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## RADIUS OF EINSTEIN-STATIC UNIVERSE FOR REALISTIC VALUES OF MATTER DENSITY

In my last blog, I considered radius of static Einstein universes with only CMB radiation of 2.7K.  We are interested in radius of static Einstein universes with realistic values for matter density.  Published estimates of matter density (for references see this) are $\rho=10^{-30} g/cm^3$ (Tipler 1987), $\rho=4-18\times 10^{-30} g/cm^3$ (Guth 1987), $\rho=5\times 10^{-30} g/cm^3$.  We need to convert to kg/m^3 so the order becomes $10^{-30} g/cm^3 = 10^{-27} kg/m^3$ and solve the equation

$\frac{8}{3c^2} \pi G \rho R^2 = 1 = k$

• $\rho=10^{-27} kg/m^3$ corresponds to a radius of 13.38 Gpc
• $\rho=5\times 10^{-27} kg/m^3$ corresponds to radius 6 Gpc
• $\rho=18\times 10^{-27} kg/m^3$ corresponds to radius 3.154 Gpc

Here we have to worry about the negative pressure issue, i.e. the second Friedmann equation for static universe is satisfied.

$\frac{\ddot{R}}{R} = -4\pi G(\rho+3p/c^2) =0$

Unlike radiation for which $\rho_{rad}+3p/c^2=0$ we don’t automatically get the cancellation for pure matter where $p=0$.  So here we need a cosmological constant to produce the negative pressure.  Here we can invoke quantum field theory for a static Einstein spacetime to obtain a cosmological constant.  The Friedmann equations with a cosmological constant are

$0 = \frac{8\pi G}{3}\rho - k c^2/R^2 + \Lambda/3$

$0 = -\frac{4\pi G}{3}(\rho + 3p) + \Lambda/3$

In the case of interest to us, where $\rho \sim 10^{-27}$ the pressure due to radiation will be much smaller, say due to CMB 2.7K $p \sim 10^{-31}$ (which is obtained by the Stefan-Boltzmann law and then dividing by $c^2$).  We fix $\rho=5 \times 10^{-27}$ and $p = 10^{-31}$ and solve for $R$ in the first equation for the sake of clarity

$kc^2/R^2 = \frac{4\pi G}{3}( 3\rho+3p)$

In this case we get the following radii for static universes:

• $\rho= 1\times 10^{-27}$ corresponds to 10.92 Gpc
• $\rho= 5\times 10^{-27}$ corresponds to 4.88 Gpc
• $\rho= 18\times 10^{-27}$ corresponds to 2.57 Gpc

This is a purely CLASSICAL picture that does not include quantum field theory effects.  When pressure $p$ increases the radii will increase as well.  For static Einstein universes the vacuum energy for example for massless neutrinos are reasonable so these could explain slightly larger radii.